Nearby Cycles and Alexander Modules of Hypersurface Complements

Abstract

Let $f:\CN \rightarrow \C $ be a polynomial map, which is transversal at infinity. Using Sabbah's specialization complex, we give a new description of the Alexander modules of the hypersurface complement $\CN\setminus f^{-1}(0)$, and obtain a general divisibility result for the associated Alexander polynomials. As a byproduct, we prove a conjecture of Maxim on the decomposition of the Cappell-Shaneson peripheral complex of the hypersurface. Moreover, as an application, we use nearby cycles to recover the mixed Hodge structure on the torsion Alexander modules, as defined by Dimca and Libgober. We also explore the relation between the generic fibre of $f$ and the nearby cycles.

comments are very welcome. arXiv admin note: text overlap with arXiv:math/0409412 by other authors

References in corpus (3)

• Milnor numbers of projective hypersurfaces with isolated singularities
• Reidemeister Torsion, Peripheral Complex, and Alexander Polynomials of Hypersurface Complements
• Some remarks on limit mixed Hodge structure and spectrum

Cited by in corpus (6)

• Reidemeister Torsion, Peripheral Complex, and Alexander Polynomials of Hypersurface Complements
• Characteristic Varieties of Hypersurface Complements
• Mixed Hodge Structures on Alexander Modules
• Hodge theory on Alexander invariants -- a survey
• Twisted Alexander Polynomials of Hypersurface Complements
• The homology groups of finite cyclic covering of line arrangement complement